In this paper, a theoretical problem arising in digital communications, namely the generalized signal richness preservation problem, is addressed and studied. In order to solve the problem, a special class of square matrices, namely the "Vandermonde-form preserving" (VFP) matrices, is introduced and found to be highly relevant to the problem. Several properties of VFP matrices are studied in detail. The necessary and sufficient conditions of the problem have been found, and a systematic proof is also presented